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(Fe$_{1-x}$Ni$_{x}$)$_{5}$GeTe$_{2}$: an antiferromagnetic triangular Ising lattice with itinerant magnetism

Based on first-principles calculations, an antiferromagnetic Ising model on a triangular lattice has been proposed to interpret the order of Fe$(1)$-Ge pairs and the formation of $\sqrt{3}$ $\times $ $\sqrt{3}$ superstructures in the Fe$_5$GeTe$_2$ (F5GT), as well as to predict the existence of similar superstructures in Ni doped F5GT (Ni-F5GT). Our study suggests that F5GT systems may be considered as a structural realization of the well known antiferromagnetic Ising model on a triangular lattice. Based on the superstructures, a Heisenberg-Landau Hamiltonian, taking into account both Heisenberg interactions and longitudinal spin fluctuations, is implemented to describe magnetism in both F5GT and Ni-F5GT. We unveil that frustrated magnetic interactions associated with Fe(1), tuned by a tiny Ni doping ($x \leq 0.2 $), is responsible for the experimentally observed enhancement of the $T_c$ to 478 K in Ni-F5GT. Our calculations show that at low doping levels, monolayer Ni-F5GT has almost the same magnetic phase diagram as that of the bulk, which indicates a pervasive beyond room temperature ferromagnetism in this Ni doped two-dimensional system.